f and f are vectors of mathematical models, 21

CANADIAN BOARD OF EXAMINERS FOR PROFESSIONAL SURVEYORS ATLANTIC PROVINCES BOARD OF EXAMINERS FOR LAND SURVEYORS ________________________________________________________________________

SCHEDULE I / ITEM 2 March 1007

LEAST SQUARES ESTIMATION AND DATA ANALYSIS

Note: This examination consists of 8 questions on 3 pages. Marks

Q. No Time: 3 hours Value Earned

1

Given the following mathematical models

2 2

1 1

0 ) ,

(

0 ) , (

2 1 2 2

1 1 1

x x

C C x

, x f

C C x

f

l l

l l

=

=

=

=

=

=

=

=

where f₁andf₂ are vectors of mathematical models, x₁and x₂ are vectors of unknown parameters, l₁andl₂ are vectors of observations,

2 1

2

1 ,C ,C and C

Cl l _{x x} are covariance matrices.

a) Linearize the mathematical models b) Formulate the variation function

10

2

Given n independent measurements of a single quantity: l₁,l₂,...,l_n and their corresponding standard deviations:σ₁,σ₂,...,σ_n, derive the expression for the variance of the mean value using the Law of

Propagation of Variances. ¹⁰

3

Given the following variance-covariance matrix from a least squares adjustment for the coordinates of point A and B,

2

x m

5 2 1 1

2 5 1 2

1 1 10 5

1 2 5 10 C

















= where x=[x_A y_A x_B y_B]^T.

a) Compute the correlation coefficient between x_A and y_B. b) Compute the correlation coefficient between y_A and x_B.

c) Compute the variance-covariance matrix for the coordinate differences

A B x x

x= −

∆ and ∆y=y_B −y_A.

10

4

Given the covariance matrix of the horizontal coordinates (x, y) of a survey station, determine the semi-major, semi-minor axis and the orientation of the standard error ellipse associated with this station.

_x m² 0196 . 0 0246 . 0

0246 . 0 0484 .

C 0 _



 



=

10

5

Define or explain the following terms:

1) Precision 2) Accuracy

3) Statistical independence 4) Type I error

5) Type II error 6) Expectation

7) Degree of freedom of a linear system

15

6

Given 20 independent measurements of a distance made with the same standard deviation of 0.033m and the calculated mean value µ= 37.615m, construct a 95% confidence interval for the population mean.

Percentiles of Standard Normal Distribution

α 0.002 0.003 0.004 0.005 0.01 0.025 0.05 Kα 2.88 2.75 2.65 2.58 2.33 1.96 1.64

where Kα is determined by the equation α= e dx 2

1 _{x /2}

K

− 2

∫∞

α π .

10

7

A baseline of calibrated length (µ) 1153.00m is measured 5 times independently with the same precision. The sample mean ( x ) and sample standard deviation (s) are calculated from the measurements:

x =1153.39m s = 0.06m

Test at a 90% confidence level (α) if the measured distance is significantly different from the calibrated distance.

Percentiles of t distribution tα

Degree of

freedom ^0.90

t t_0.95 t_0.975 t_0.99

1 3.08 6.31 12.7 31.8

2 1.89 2.92 4.30 6.96

3 1.64 2.35 3.18 4.54

4 1.53 2.13 2.78 3.75

5 1.48 2.01 2.57 3.36

10

8

Given in the following is a diagram of a leveling network to be adjusted.

Table 1 contains all of the height difference measurements ∆h_ij necessary to carry out the adjustment. Assume that all observations have the same standard deviation

hij

σ∆ = 0.04m. P1 is a fixed point with known elevations of H1 = 101.00m. Perform least squares adjustment on the leveling network to compute:

a) the adjusted elevations of point P2, P3 and P4 and their variance- covariance matrix;

b) the observation residuals;

c) the adjusted height differences; and d) the a-posteriori variance factor σˆ₀. Table 1

No. i j ∆h_ij(unit: metre)

1 P1 P3 6.15

2 P1 P4 12.54

3 P3 P4 6.46

4 P1 P2 1.11

5 P2 P4 11.55

6 P2 P3 5.10

25

Total Marks: 100 P1

P2

P3 P4